- Instructor name: Pál Zsámboki
- Instructor email: pzsambok@uwo.ca
- Instructor office: MC 133
- Class webpage: on OWL
- Documents on OWL page:
- Class Outline written by the Course Coordinator
- List of practice problems

- Textbook: Single Variable Calculus: Early Transcendentals (8th edition), by James Stewart (Brooks/Cole)
- It is important that you have access to the correct edition, since there is a list of practice problems we'll suppose you know how to solve.

- Office hours: Thursdays 3pm-5pm. No office hour on September 23.
- Math Help Centre: TBD, starting on September 19

- Midterm exam 35%
- The midterm exam will be on Friday October 21, from 7pm-9pm. Location TBA. The exam will be closed book, no notes, no calculators or any other electronic devices will be permitted.

- Final exam 50%
- The final exam will be cumultative, and it will be 3 hours long. Location TBA. The exam will be closed book, no notes, no calculators or any other electronic devices will be permitted.

- Quizzes 15%
- There will be 6 online quizzes on OWL, with 2 questions in each.
- You will have 2 weeks for every one of them, and you'll have 3 tries with each.
- Each quiz will have a time limit of 20 minutes.
- Your worst quiz grade will be dropped.
- For more details, check quizzes.txt.

- I think the most intuitive way of thinking about trigonometric functions is that the function \(f(\theta)=(\cos(\theta),\sin(\theta))\) is the
*parametrization*of the*unit circle*with respect to*arc length*, in*counterclockwise*or in other words*positive direction*, starting from the point \(f(0)=(1,0)\).- That is, the function \(f(\theta)\) traces out the unit circle as \(\theta\) varies
- Being parametrized with respect to arc length means that if we change \(\theta\) by 1 unit, the parametrization \(f(\theta)\) traces out an arc of length 1.

- The standard parametrization \(f(\theta)=(\cos(\theta),\sin(\theta))\) corresponds to the following conventions.
- An angle is said to be in
*standard position*, if its vertex is placed at the origin of the coordinate system we're using, and its initial side is on the positive \(x\)-axis - A
*positive angle*is obtained if going from the initial angle to the terminal angle is a counterclockwise rotation - A
*negative angle*is obtained if going from the initial angle to the terminal angle is a clockwise rotation

- An angle is said to be in

- If the angle \(\theta\) is measured like this, that is in the way that the parametrization of the unit circle \(f(\theta)=(\cos(\theta),\sin(\theta))\) is with respect to arc length, we say that it is measured in
*radians*. Since the circumference of the unit circle is \(2\pi\), a complete revolution is done by changing \(\theta\) by \(2\pi\).- Let \(r>0\) be a positive number. Then similarly we can parametrize the circle \(C\) with centre the origin and radius \(r\) via \(f(\theta)=(r\cos(\theta),r\sin(\theta))\).
- Note that this parametrization is not with respect to arc length: a complete revolution around \(C\), which traces out a path of length the circumference of \(C\), that is \(a=2\pi r\), corresponds to a change of \(2\pi\) in \(\theta\).
- More generally, for a circular arc of length \(a\), radius \(r\), and angle \(\theta\), we get the formula \(a=\theta r\).

- There is another, historical angle unit, that of
*degrees*. If angles are measured in degrees, then a complete revolution is \(360^\circ\).- This gives us the conversion formula between radians and degrees: \(2\pi=360^\circ\).
- A related unit of length which I'm rather fond of is that of
*nautical miles (NM)*. 1 nautical mile is 1 latidual degree minute. That is, if you travel 60 NM from North to South, then you travelled precisely \(\tfrac{1}{360}\) of the latitudal circumference of the Earth. - Exercises: D.2,4,8,10

- Consider a right triangle \(T\), where one of the non-right angles is \(\theta\).
- Suppose that the angle \(\theta\) is in standard position, let's give it the positive orientation, and let's denote the length of the hypotenuse by \(r\).
- Then we can study the triangle \(T\) with the parametrization \(f(\theta)=(x=r\cos(\theta),y=r\sin(\theta))\).
- Its vertices are going to be \(O(0,0)\), \(X(x,0)\), and \(P(x,y)\).
- The side \(\overline{OX}\) is the
*adjacent*, the side \(\overline{XP}\) is the*opposite*, and the side \(\overline{OP}\) is the*hypotenuse*. - The lengths of the legs are \(|OX|=x\), and \(|XP|=y\).
- Then we can express the trigonometric functions using these side lengths:
- \(\cos(\theta)=\frac{x}{r}\), \(\sin(\theta)=\frac{y}{r}\), \(\tan(\theta)=\frac{y}{x}=\frac{\sin(\theta)}{\cos(\theta)}\),
- \(\sec(\theta)=\frac{r}{x}=\frac{1}{\cos(\theta)}\), \(\csc(\theta)=\frac{r}{y}=\frac{1}{\sin(\theta)}\), \(\cot(\theta)=\frac{x}{y}=\frac{\cos(\theta)}{\sin(\theta)}\)
- Since the parametrization \(f(\theta)\) is defined for all values of \(\theta\), we can use the same formulas in case \(\theta<0\) or \(\theta\ge\frac{\pi}{2}\).

- Note the following properties on the graphs of \(\cos(x)\) and \(\sin(x)\).
- Both functions are \(2\pi\)-periodic, their domain is \((-\infty,\infty)\), and their range is \([-1,1]\).
- \(\cos(x)=\sin(x+\frac{\pi}{2})\). This can be also seen by rotating the right triangle \(OXP\) by \(\frac{\pi}{2}\).
- \(\cos(x)=0\) iff (if and only if) \(x=\frac{\pi}{2}+k\pi\) for some integer \(k\), and \(\sin(x)=0\) iff \(x=k\pi\).
- I think it makes remembering the special values easier to notice that on the interval \([0,\frac{\pi}{2}]\), the functions \(\cos(x)\) is decreasing, and the function \(\sin(x)\) is increasing.